The Eulerian distribution on involutions is indeed unimodal

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Abstract

Let In, k (respectively Jn, k) be the number of involutions (respectively fixed-point free involutions) of {1, ..., n} with k descents. Motivated by Brenti's conjecture which states that the sequence In, 0, In, 1, ..., In, n - 1 is log-concave, we prove that the two sequences In, k and J2 n, k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an, k such thatunderover(∑, k = 0, n - 1) In, k tk = underover(∑, k = 0, ⌊ (n - 1) / 2 ⌋) an, k tk (1 + t)n - 2 k - 1 . This statement is stronger than the unimodality of In, k but is also interesting in its own right. © 2005 Elsevier Inc. All rights reserved.

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Guo, V. J. W., & Zeng, J. (2006). The Eulerian distribution on involutions is indeed unimodal. Journal of Combinatorial Theory. Series A, 113(6), 1061–1071. https://doi.org/10.1016/j.jcta.2005.10.002

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